Basic equivalence relation for sd-keylist structures.
Function:
(defun sd-keylist-equiv$inline (acl2::x acl2::y) (declare (xargs :guard (and (sd-keylist-p acl2::x) (sd-keylist-p acl2::y)))) (equal (sd-keylist-fix acl2::x) (sd-keylist-fix acl2::y)))
Theorem:
(defthm sd-keylist-equiv-is-an-equivalence (and (booleanp (sd-keylist-equiv x y)) (sd-keylist-equiv x x) (implies (sd-keylist-equiv x y) (sd-keylist-equiv y x)) (implies (and (sd-keylist-equiv x y) (sd-keylist-equiv y z)) (sd-keylist-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm sd-keylist-equiv-implies-equal-sd-keylist-fix-1 (implies (sd-keylist-equiv acl2::x x-equiv) (equal (sd-keylist-fix acl2::x) (sd-keylist-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm sd-keylist-fix-under-sd-keylist-equiv (sd-keylist-equiv (sd-keylist-fix acl2::x) acl2::x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-sd-keylist-fix-1-forward-to-sd-keylist-equiv (implies (equal (sd-keylist-fix acl2::x) acl2::y) (sd-keylist-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-sd-keylist-fix-2-forward-to-sd-keylist-equiv (implies (equal acl2::x (sd-keylist-fix acl2::y)) (sd-keylist-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm sd-keylist-equiv-of-sd-keylist-fix-1-forward (implies (sd-keylist-equiv (sd-keylist-fix acl2::x) acl2::y) (sd-keylist-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm sd-keylist-equiv-of-sd-keylist-fix-2-forward (implies (sd-keylist-equiv acl2::x (sd-keylist-fix acl2::y)) (sd-keylist-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)